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There are a number of places to see that are not too far away. Lago di Garda and the towns around it such as Sirmione, Riva del Garda, etc, are great. Verona is not far away.
Bassano del Grappa, Vicenza and Brescia are a bit farther but are all nice to see. Venice is a bit farther but can be reached by train in 2.5 hours from Trento so it still can be done as a day trip. Milan is under 3 hours by train. Driving through the Dolimites toward Belluna is also a nice drive (lots and lots of hairpins) but making it the whole way and back in a day is asking a lot of the driver.

So there are a lot of things that are well worth seeing that are not too far away. Certainly, you could stay a week and not do all of the things that I have mentioned above.

Verona is closer to fly into but the extra connections may not be worth it.

I can only suggest you take SS12 straight south from Trento to Verona. It is an awesome drive. Plan on stopping in Riva Del Garda for a dinner on your way back.

Stop at Rovereto at some point. It's only 20-30 min south of Trento on A22 Autostrada and my wife loved that quaint little town. I have a very good friend living there so I visited that town at least 20 times. PM me if you want best restaurant in town suggestions - I'll ask my friend for both Trento and Rovereto.

If you want to eat best risotto money can buy, Isola Della Scala is your town, just a tad south from Verona. I mean it, it is worth a trip from Trento. It's in the heart of the area where most of Italian rice is grown, so you will be eating local stuff.